Gradient estimate and Universal bounds for semilinear elliptic equations on RCD$^*$(K,N) metric measure spaces

Abstract

We derive logarithmic gradient estimate and universal boundedness estimate for semilinear elliptic equations on \RCD\, metric measure spaces, which contains the class of Riemannian manifolds with Ricci curvature bounded below. These estimates are applicable for equations satisfying subcritical index condition,which recover many classical results even on Euclidean spaces. In certain case, these estimates are optimal even on \RCD\,\,spaces with $K<0$. Two direct corollaries of these estimates are Harnack inequality and Liouville theorem. In addition to these estimates, we also establish fundamental relations among the universal boundedness estimate, the logarithmic gradient estimate, and Harnack inequality. Under certain and wild assumptions for the nonlinear term, we prove that these estimates are $κ$-equivalent on \RCDO\,spaces for any $κ>1$.